Plot distance vs. velocity. We find:
•Velocity proportional to distance
v  H0d
•H0 is called Hubble’s constant
•Best fit today (includes many assumptions):
H 0  70.4  1.4 km/s/Mpc
Uncertainties:
•Until recently, we didn’t know H0
very well, so we would write
•h = 0.704  0.014
H 0  100h km/s/Mpc
•I’ll probably avoid h.
•Many published documents
write answers in terms of h so that
b h 2  0.02260  0.00053
we remove the uncertainty in H0
Distances from Hubble’s Law:
•For almost any bright object, you can get red shift z:
•You can then get the velocity from
1 v c
v
•Then use Hubble’s Law:
1 z 
 1
1 v c
c

1 z 
0
v  H0d
A spectral line from a galaxy normally at 415.0 nm appears instead at 423.0 nm.
(a) How fast is the galaxy moving away from us?
(b) What is the distance to the galaxy?

423.0
v
1 z 

 1.0193  1 
0 415.0
c
d
v
5790 km/s

H0
70.4 km/s/Mpc
v  0.0193c
d  82 Mpc
v  5790 km/s
Peculiar Velocities
Hubble’s Law isn’t perfect: low z:
•Not everything expands
•Galaxies, galaxy clusters, and smaller objects are not necessarily expanding
•Galaxy superclusters expand more slowly than Hubble’s Law implies
•Objects have extra velocities, called peculiar velocities
•Because gravity changes velocities
•Typically of order 500 km/s or so
•This causes errors, usually random of vp /H0 in the distance
•Typically 7 Mpc or so
•Poor distance indicator for distances under 50 Mpc or so.
•We also have our own peculiar velocity that needs to be compensated for
•We know how to do this
Failure of Hubble’s Law at High z:
Another reason it fails: high z:
•As z gets large (> 0.2) objects are moving relativistically
•Relativistic corrections
•Distance gets confusing as well
•We are also looking back into the past
•Bad: Expansion might not be constant
•Good: We can study the universe in the past (!)
•Still, higher z corresponds to greater distance
•If we can accurately measure distance to some high z objects, we can set up a
calibration scale
The Cosmic Distance Ladder:
Hubble’s Law
Cepheid Variables
Radar
Parallax
Planetary Nebulas
Moving
Cluster
Type Ia Supernovae
Light
Echo
Spectroscopic Parallax
Cluster Fitting
1 AU
1
10
pc
100
1
10
kpc
100
1
10
100
Mpc
1
10
Gpc
Hubble Expansion
Hubble’s Law can be thought of two ways:
•All galaxies are flying apart from each other
•The space between the galaxies is expanding
There is no
special place in
the universe
•It is meaningless
to ask “where is it
expanding from”
•All observers see
the same thing
Age of the Universe: The naïve view:
•Hubble’s Law tells us relation between distance and velocity
•We can therefore figure out how long ago all this stuff was here:
•Call t0 the time when everything left here:
d  vt0
d  H 0 dt0
H 0 t0  1
t0  H 01
1
H
•This time is called the inverse Hubble time: 0  13.9  0.3 Gyr
•This is incorrect because the velocity of other
galaxies is probably not constant
Assuming gravity slows things down, should the actual age of
the universe be grater than the inverse Hubble time, or less?
•Speeds are slowing down
•Speeds used to be greater
•Higher speed  Less time to get to where we are now
v  H0d
The Friedman Equation (1)
•Assume the universe is homogenous and isotropic
•We can treat any point (us), as center of the universe
a
•We can use spherical symmetry around us
distant
•Let a be the distance to any specific distant galaxy
us
galaxy
•And let its mass be m
M
•By Gauss’s Law, gravitational force is easy to find
•Spherical symmetry tells us force is towards center
d 2a
GMm
•And caused by only mass closer than that galaxy
F  2 m 2
dt
a
•Because expansion of the universe is uniform, stuff
2
d
a
GM
closer than galaxy always closer
 2
2
dt
a
•M is constant
d 2 a da
2GM da
•Multiply both sides by 2da/dt and integrate over time
2
dt


dt
2
2


Don’t forget the constant of integration!
dt dt
a dt
2
•We could call it k, for example
2GM
 da 
2
2

•For technical/historical reasons, it is called – kc

kc
 
a
 dt 
•This makes k dimensionless
The Friedmann Equation (2)
2
•Mathematical notation: Time derivatives  da 
2GM
2


kc
 
a
2GM
 dt 
a
2
2
da
a


kc
a
a
dt
a 2 2GM kc 2
distant
us
 3  2
2
galaxy
a
a
a
•Divide both sides by a2
M
•Rewrite first term on right in
3
4
M


V


a

3
terms of mass density 
2
2
a
kc
The resulting equation is called the First Friedman Equation
8

3  G 
2
a
a2
•The derivation was flawed
•M does not necessarily remain constant
a
•In relativity, other contributions to gravity
H
a
•The equation is correct, including relativity
a  Ha
•The left side is the square of Hubble’s constant
•Left side and first term on right independent of which galaxy you use
•We can pick a particular galaxy to make k bigger or smaller
•Normally chosen so that k = 0, +1, or -1
The Scale Factor
k  1, 0, or  1
a2 8
kc 2
 3  G  2
2
a
a
a
H
a
•The distance a is called the scale factor
•All distances increase with the same ratio
•This applies to anything that is not bound together
•Even to waves of light!
Red shift can be thought of as a stretching of universe
•Or is it Doppler shift?
•Either view is correct, there isn’t a right answer
•This view does give us another relationship:
a then
a
1
d
1




a0 1  z d 0
a0 now 1  z
•Most common way to label something in the distant past: z
•Bigger z means earlier
a
d

a0 d 0
Early a
Universe
Now
a0

k  1, 0, or  1
a2 8
kc 2
 3  G  2
2
a
a
a
H
a
•We’d like to know which value of k is right
8
3  G
•Define the ratio:

H2
•If we know how big  is, we’ll know k
k
2
2
2
2
H  H  2
k
a

H
  1
a
8
•It is common to break  into pieces
3  G i
i 
•Ordinary atoms and stuff: b
H2
•Dark Matter: d
   i
•Matter: m = d + b
i
•Radiation: r
•Etc.
• changes with time
•When referring to values today, we add the subscript 0
•Sometimes
Dens.
<1
=1
>1
Curv.
k = -1
k=0
k = +1
  t0   0
Name
Open
Flat
Closed
Curvature
•What’s the circumference of a circle of radius r?
•According to Einstein, Space time can be curved!
•Shape of space depends on density parameter: 
•If  = 1 then universe is flat and
C  2 r
•If  > 1 then universe is closed and
C  2 a sin  r a 
•If  < 1 then universe is open and
C  2 a sinh  r a 
•Surface area of a sphere:

4 r 2
  1,

A   4 a 2 sin 2  r a    1,
4 a 2 sinh 2  r a    1.

•A closed universe is finite
Dens.
<1
=1
>1
Curv.
k = -1
k=0
k = +1
Name
Open
Flat
Closed
So, What is ?
Stuff
Stars
Gas, Dust
Neutrinos
Dark Matter
Light
Total
Cont. to 
0.011
b = 0.0456  0.0016
0.035
10-4 – 10-2
0.227  0.014
~ 10-5
~ 0.273
As we will see, we have missed a substantial fraction of 
Age of the Universe, Round 2 (page 1)
•What’s the current density of stuff in the universe?  
8
3
 G a 2
2
kc
8

3  G 
2
a
a2
H2
2
kc
 G 0  H 0
 2  H 02 1  0 
a0
•For ordinary matter, density  1/Volume
8
3
2
0
3
8
3
 a0 
 a0 
2

H

0

a
a
 
 
3
 G   83  G 0 
 a0 
kc
kc  a0 
2
 2   2    H 0 1     
a
a0  a 
 a
2
2
2
3
•Substitute in:
2
a2
 a0 
 a0 
2
2
 H 0     H 0 1     
2
a
a
a
2
Age of the Universe, Round 2 (page 2)
3
a2
 a0 
 a0 
2
2
 H 0     H 0 1     
2
a
a
a
•Let x = a/a0 :
•If  = 0.273, then
•t0 = 11.4  0.3 Gyr
•If  = 1.000, then
•t0 = 9.3  0.2 Gyr
•These are younger than
the oldest stars
The Age Problem
2
x2
2
3
2
2

H

x

H
1


x


0
0
x2

x 2  H 0 x 1  1  

x  H0  x  1  
dt
1
1
 H0
dx
 x 1 
t0  H
1
0
1

0
dx
 x 1 